GCSE Maths

Numbers
Vectors
Probability
Statistics
Algebra
Sequences

To solve the quadratic equation , we factorise the quadratic expression into two factors. The solutions are then obtained by setting each of the factors equal to zero. We shall obtain two solutions.

Let’s look at some examples.

Example

Solve the equations:

i)

ii)

i)

Either

Or,

ii)

So, either

Or,

Example

Or, we can do this by factors

Either

Or,

Example

Solve the equation

We multiply both sides by x.

Then, we expand the bracket and simplify.

Factorise:

Solve: or

The quadratic expression is not always factorisable in rational numbers. This situation arises when the quantity , called the discriminant, is not a perfect square, meaning it has no rational square roots.

We then solve the quadratic equation , using the formula

For the equation:

, and .

We substitute these numbers into the formula above. So:

We obtain

(3 significant figures)

Or,

(3 significant figures)

Let’s look at some examples.

Example

Giving answers correct to three significant figures

or , each correct to three significant figures.

Example

Find the possible values of x for which:

Here, , and .

We substitute these numbers into

Or,

The possible values of x are and 3.

Note: In fact, the quadratic equation can be solved using factorisation.

Either , giving

Or, , giving .

Example

A solid cuboid of length x metres has a square cross-section of side metres. Write down, in terms of x, expressions for:

i) The surface are of the square cross-section.

ii) The surface area of one of the longer faces of the cuboid.

Given that the total surface area of the cuboid is 39 square metres, write down an equation in x and show that it reduces to:

Find the two values of x which satisfies this equation, giving your answers correct to three significant figures. Hence right down the dimensions of the cuboid.

i) Surface are of the cross-section

ii) Surface area of one of the longer faces

Total surface area of the cuboid:

Given the total surface area :

Dividing by the common factor 3, we obtain:

, as required.

We use the formula to find x

or , each to three significant figures.

The dimensions of the cuboid are 4.74 m, 1.74 m

Example

Solve the equation:

We can simplify the left hand side, then cross-multiply.

So,

We now cross-multiply and expand:

Transposing:

Either

or

= 1.54

to three significant figures.

Example – Full exam question

(a) Find the value of:

When and , give your answer as a fraction in its lowest term.

(b) Expand the brackets and simplify:

(c)(i) Factorise

(d) The sum of three consecutive even numbers is 78. Find these three numbers.

(a)

When , , we obtain:

(b)

(c)(i)

Find two numbers, sum = 5, product= 36.

1┃36

2┃18

3┃12

4┃9 → Good, choose 4,9.

(ii)

Either

or

(d) Let the three even numbers be:

, and , as they are consecutive.

Now,

So that the three even numbers are:

Example – Exam question

The distance between London and York is 320 km. A train takes x hours to travel between London and York.

(a) Write down an expression, in terms of x, for the average speed of the train in km/h.

(b) A car takes 2 \frac{1}{2} hours longer than a train to travel between London and York.

The average speed of the train is 80 km/h greater than the average speed of the car.

Form an equation in x and show that it simplifies to .

(c) Solve the equation , giving your answer correct to two decimal places.

(d) Hence find the average speed of the car correct to the nearest km/h.

(a) London → York = 320 km

Average speed

Average speed of the train

(b) Average speed of the car

Average speed of the train average speed of the car = 80

Divide by 80 throughout the equation:

Multiply by on both sides

Multiply by 2 on both sides

Rearranging gives

(c) Solving

Either

(two decimal places)

Or

(two decimal places)

(d) Average speed of the car

km/h

69 km/h (nearest km/h)

Example

Solve the equation:

We first need to rewrite the equation into the standard form

Either

Or